2 - Regular Polytopes

Summary

It is not an easy task to trace precisely the history of the idea of abstract regular polytopes. While it is clear that the concept has its roots in the classical theory, and notably in Coxeter's work [120], more recently there have been several parallel developments which have influenced the theory of abstract regular polytopes.

From the point of view of discrete geometry, it appears that combinatorial regularity was first studied in McMullen [277] in the context of combinatorially regular convex polytopes (see Section 1B). In its generality, the notion of an abstract regular polytope was largely anticipated in Grünbaum's paper [199] on structures which he called regular polystromata. Then, in 1977, Danzer introduced the more restrictive concept, based on Grünbaum's work, of a regular incidence complex; see Danzer and Schulte [141], although the definitions adopted were anticipated by McMullen (in a geometric context) in [280, p. 578]. Among these regular incidence complexes, the abstract regular polytopes, or regular incidence polytopes as they were first called, are particularly close to the traditional polytopes, and form a special class of polytope-like structures with a distinctive geometric and topological appeal. It seems that a more systematic study of these objects was begun by Schulte [362–364].